VPN identification question

I have a question regarding VPNs and potential identification. I was unable to find an answer to my specific question (or maybe I was unable to formulate it properly). Here it goes:

Say I use a VPN that obfuscates my real IP address. Although my ISP will only be able to see encrypted data, it will still be able to see my VPN-IP address, right? So I assume this VPN-IP address is now logged by the ISP. The ISP can now connect this VPN-IP address with my person, right?
Let’s say I visit server/website X or download via torrent. I assume my VPN-IP address is now logged by server/website X.

Let’s theoretically assume that authorities are interested in finding out who really is behind my VPN-IP address because that IP address has done something of interest and they got the VPN-IP address by going through the logs of server/website X (which I visited earlier).
Wouldn’t it be possible for authorities to identify me by contacting major ISPs and asking them which clients have caused traffic coming from that IP address.

That question has interested me for some time. Maybe someone can finally shed some light on it.

Correct. This is why we strongly discourage the use of a VPN for anonymity purposes.

Yes, you can be identified, not a big task. Moreover if the vpn provider is from US, then they need to submit any records about the client under the PRISM surveillance, without any subpoena.

This might interest you::

Hey, got this info.